Over the past four years, the islands' population (17,500) has resoundingly defeated two referendums to become an independent country. Residents hold steadfastly to both their political and cultural ties with the United Kingdom.

The Union Jack flies prominently over all official buildings. The shops are full of Prince Charles and Princess Diana memorabilia, from coffee mugs to sterling napkin rings. Schoolchildren sport the tidy uniforms and follow the rigorous academic courses of their British counterparts.

''It's not that we are unaware of the possibility of independence,'' observes a Cayman Islands official, ''it's simply that the people like being part of England.''

The population is a broad mix of races and influences, Scottish and blacks most prominently. Britain abolished slavery in 1835, and since then the various races have intermarried freely. Perhaps most strikingly, there is no sense of brooding discontent beneath the surface of the islands' normal activities.

Says a young Caymanian, ''I lived in the Bahamas, but I prefer it here. It's much freer and there is not color prejudice the way there is in the Bahamas.''

Financial stability is the most obvious asset of such tranquillity. The Caymans have witnessed an explosion in financial activity over the last 20 years. In 1966, the islands had three banks; now they have more than 400.

There are no personal, corporate, profit, capital gains, property, or inheritance taxes. The only direct taxation is a yearly $10 head tax on males between the ages of 18 and 60. The islands are, in short, a financial haven for foreign investment.

Two events in 1966 precipitated this status. First, renovation of airport runways to accommodate jet landings and second, assembly legislation that formalized their tax haven status.

Notices such as one in a bank lobby stating, ''We take no pictures,'' feed the image that Cayman banks do not care whether their customers are depositing hot cash. But according to a United States investment counselor who has worked extensively in the Caymans, most banks are not even equipped to accept cash deposits. In the time he worked on the islands, he never accepted one. He says the banks go to great lengths to emphasize the legality of what they offer.

Caymanian tourism received a boost when nearby Jamaica's political upheaval sent tourists looking for a friendlier vacation spot. And development on the islands continues at a rapid pace despite restrictions such as a seven-story limit on new buildings.

''The Caymans have changed enormously in just the last few years,'' notes one resident. ''People want a retreat that is really a retreat. The Caymans still offer such a place, but I don't know for how much longer.''